Optimal State-Space Order for Spectral Gaps of Sliding-Window Occupation Counts

Let $P$ be an irreducible reversible Markov kernel on a $m$-state space $Ω$, and denote its right spectral gap $γ=1-λ_2(P)$. From a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible pro